4.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


4.1A 
Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

Apply
MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:
 Mathematical problem situations within and between disciplines
 Everyday life
 Society
 Workplace
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Measuring angles
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 TxCCRS:
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld situation.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
 IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.

4.1B 
Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.
Process Standard

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving process and the reasonableness of the solution
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Measuring angles
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.A. Statistical Reasoning – Design a study
 V.A.1. Formulate a statistical question, plan an investigation, and collect data.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VII.A.2. Formulate a plan or strategy.
 VII.A.3. Determine a solution.
 VII.A.4. Justify the solution.
 VII.A.5. Evaluate the problemsolving process.
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.2. Evaluate the problemsolving process.

4.1C 
Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.
Process Standard

Select
TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:
 Appropriate selection of tool(s) and techniques to apply in order to solve problems
 Tools
 Real objects
 Manipulatives
 Paper and pencil
 Technology
 Techniques
 Mental math
 Estimation
 Number sense
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Measuring angles
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.

4.1D 
Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Process Standard

Communicate
MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:
 Mathematical ideas, reasoning, and their implications
 Multiple representations, as appropriate
 Symbols
 Diagrams
 Graphs
 Language
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Measuring angles
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 TxCCRS:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.

4.1E 
Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

Create, Use
REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:
 Representations of mathematical ideas
 Organize
 Record
 Communicate
 Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
 Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Measuring angles
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 TxCCRS:
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.

4.1F 
Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

Analyze
MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:
 Mathematical relationships
 Connect and communicate mathematical ideas
 Conjectures and generalizations from sets of examples and nonexamples, patterns, etc.
 Current knowledge to new learning
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Measuring angles
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 TxCCRS:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.

4.1G 
Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

Display, Explain, Justify
MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION
Including, but not limited to:
 Mathematical ideas and arguments
 Validation of conclusions
 Displays to make work visible to others
 Diagrams, visual aids, written work, etc.
 Explanations and justifications
 Precise mathematical language in written or oral communication
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Measuring angles
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 TxCCRS:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.4. Justify the solution.
 VII.B. Problem Solving and Reasoning – Proportional reasoning
 VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
 VII.C. Problem Solving and Reasoning – Logical reasoning
 VII.C.1. Develop and evaluate convincing arguments.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.

4.2 
Number and operations. The student applies mathematical process standards to represent, compare, and order whole numbers and decimals and understand relationships related to place value. The student is expected to:


4.2A 
Interpret the value of each placevalue position as 10 times the position to the right and as onetenth of the value of the place to its left.
Supporting Standard

Interpret
THE VALUE OF EACH PLACEVALUE POSITION AS 10 TIMES THE POSITION TO THE RIGHT AND AS ONETENTH OF THE VALUE OF THE PLACE TO ITS LEFT
Including, but not limited to:
 Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, ten thousands, etc.
 One billions place
 Hundred millions place
 Ten millions place
 One millions place
 Hundred thousands place
 Ten thousands place
 One thousands place
 Hundreds place
 Tens place
 Ones place
 Tenths place
 Hundredths place
 Base10 place value system
 A number system using ten digits 0 – 9
 Relationships between places are based on multiples of 10.
 Moving left across the places, the values are 10 times the position to the right.
 Moving right across the places, the values are onetenth the value of the place to the left.
 Place value relationships and relationships between the values of digits involving whole numbers through (less than or equal to) 1,000,000,000 and decimals to the hundredths (greater than or equal to 0.01)
 Place value relationships are based on multiples of ten whereas relationships between the values of digits may or may not be based on multiples of ten.
Note(s):
 Grade Level(s):
 Grade 3 described the mathematical relationships found in the base10 place value system through the hundred thousands place.
 Grade 5 will represent the value of the digit in decimals through the thousandths using expanded notation and numerals.
 Various mathematical process standards will be applied to this student expectation as appropriate .
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.2. Interpret the relationships between the different representations of numbers.

4.2B 
Represent the value of the digit in whole numbers through 1,000,000,000 and decimals to the hundredths using expanded notation and numerals.
Readiness Standard

Represent
THE VALUE OF THE DIGIT IN WHOLE NUMBERS THROUGH 1,000,000,000 AND DECIMALS TO THE HUNDREDTHS USING EXPANDED NOTATION AND NUMERALS
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
 Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
 Decimals (great than or equal to 0.01)
 Decimal number – a number in the base10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
 Numeral – a symbol used to name a number
 Digit – any numeral from 0 – 9
 Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, ten thousands, etc.
 One billions place
 Hundred millions place
 Ten millions place
 One millions place
 Hundred thousands place
 Ten thousands place
 One thousands place
 Hundreds place
 Tens place
 Ones place
 Tenths place
 Hundredths place
 Base10 place value system
 A number system using ten digits 0 – 9
 Relationships between places are based on multiples of 10.
 Moving left across the places, the values are 10 times the position to the right.
 Multiplying a number by 10 increases the place value of each digit.
 Moving right across the places, the values are onetenth the value of the place to the left.
 Dividing a number by 10 decreases the place value of each digit.
 The magnitude (relative size) of whole number places through the billions place
 The magnitude of one billion
 1,000,000,000 can be represented as 10 hundred millions.
 1,000,000,000 can be represented as 100 ten millions.
 1,000,000,000 can be represented as 1,000 one millions.
 The magnitude (relative size) of decimal places through the hundredths
 The magnitude of onetenth
 0.1 can be represented as 1 tenth.
 0.1 can be represented as 10 hundredths.
 The magnitude of onehundredth
 0.01 can be represented as 1 hundredth.
 Standard form – the representation of a number using digits (e.g., 985,156,789.78)
 Period – a threedigit grouping of whole numbers where each grouping is composed of a ones place, a tens place, and a hundreds place, and each grouping is separated by a comma
 Billions period is composed of the one billions place, ten billions place, and hundred billions place.
 Millions period is composed of the one millions place, ten millions place, and hundred millions place.
 Thousands period is composed of the one thousands place, ten thousands place, and hundred thousands place.
 Units period is composed of the ones place, tens place, and hundreds place.
 The word “billion” after the numerical value of the billions period is stated when read.
 A comma between the billions period and the millions period is recorded when written but not stated when read.
 The word “million” after the numerical value of the millions period is stated when read.
 A comma between the millions period and the thousands period is recorded when written but not stated when read.
 The word “thousand” after the numerical value of the thousands period is stated when read.
 A comma between the thousands period and the units period is recorded when written but not stated when read.
 The word “unit” after the numerical value of the units period is not stated when read.
 The word “hundred” in each period is stated when read.
 The words “ten” and “one” in each period are not stated when read.
 The tens place digit and ones place digit in each period are stated as a twodigit number when read.
 The whole part of a decimal number is recorded to the left of the decimal point when written and stated as a whole number.
 The decimal point is recorded to separate the whole part of a decimal number from the fractional part of a decimal number when written and is stated as “and” when read.
 The fractional part of a decimal number is recorded to the right of the decimal point when written.
 The fractional part of a decimal number is stated as a whole number with the label of the smallest decimal place value when read (e.g., 0.5 is read as 5 tenths; 0.25 is read as 25 hundredths; etc.).
 The “ths” ending denotes the fractional part of a decimal number.
 Zeros are used as place holders between digits of a number as needed, whole part and fractional part, to maintain the value of each digit (e.g., 400.05).
 Leading zeros in a decimal number are not commonly used in standard form, but are not incorrect and do not change the value of the decimal number (e.g., 0,037,564,215.55 equals 37,564,215.55).
 Trailing zeros after a fractional part of a decimal number may or may not be used and do not change the value of the decimal number (e.g., 400.50 equals 400.5).
 Word form – the representation of a number using written words (e.g., 985,156,789.78 as nine hundred eightyfive million, one hundred fiftysix thousand, seven hundred eightynine and seventyeight hundredths)
 The word “billion” after the numerical value of the billions period is stated when read and recorded when written.
 A comma between the billions period and the millions period is not stated when read but is recorded when written.
 The word “million” after the numerical value of the millions period is stated when read and recorded when written.
 A comma between the millions period and the thousands period is not stated when read but is recorded when written.
 The word “thousand” after the numerical value of the thousands period is stated when read and recorded when written.
 A comma between the thousands period and the units period is not stated when read but is recorded when written.
 The word “unit” after the numerical value of the units period is not stated when read and not recorded when written.
 The word “hundred” in each period is stated when read and recorded when written.
 The words “ten” and “one” in each period are not stated when read and not recorded when written.
 The tens place digit and ones place digit in each period are stated as a twodigit number when read and recorded using a hyphen, where appropriate, when written (e.g., twentythree, thirteen, etc.).
 The whole part of a decimal number is recorded the same as a whole number with all appropriate unit labels prior to recording the fractional part of a decimal number.
 The decimal point is recorded as the word “and” to separate the whole part of a decimal number from the fractional part of a decimal number when written and is stated as “and” when read.
 The fractional part of a decimal number followed by the label of the smallest decimal place value is recorded when written and stated when read.
 The “ths” ending denotes the fractional part of a decimal number.
 The zeros in a number are not stated when read and are not recorded when written (e.g., 854,091,005.26 in standard form is read and written as eight hundred fiftyfour million, ninetyone thousand, five and twentysix hundredths in word form).
 Place Value forms
 Expanded form – the representation of a number as a sum of place values (e.g., 985,156,789.78 as 900,000,000 + 80,000,000 + 5,000,000 + 100,000 + 50,000 + 6,000 + 700 + 80 + 9 + 0.7 + 0.08)
 Zero may or may not be written as an addend to represent the digit 0 in a number (e.g., 905,150,089.08 as 900,000,000 + 0 + 5,000,000 + 100,000 + 50,000 + 0 + 0 + 80 + 9 + 0.0 + 0.08 or as 900,000,000 + 5,000,000 + 100,000 + 50,000 + 80 + 9 + 0.08).
 Expanded form is written following the order of place value.
 The sum of place values written in random order is an expression but not expanded form.
 Expanded notation – the representation of a number as a sum of place values where each term is shown as a digit(s) times its place value (e.g., 985,156,789.78 as 9(100,000,000) + 8(10,000,000) + 5(1,000,000) + 1(100,000) + 5(10,000) + 6(1,000) + 7(100) + 8(10) + 9 + 7(0.1) + 8(0.01) or 985,156,789.78 as 9(100,000,000) + 8(10,000,000) + 5(1,000,000) + 1(100,000) + 5(10,000) + 6(1,000) + 7(100) + 8(10) + 9 + 7() + 8())
 Zero may or may not be written as an addend to represent the digit 0 in a number (e.g., 905,150,089.08 as (9 × 100,000,000) + (0 × 10,000,000) + (5 × 1,000,000) + (1 × 100,000) + (5 × 10,000) + (0 × 1,000) + (0 × 100) + (8 × 10) + (9 × 1) + (0 × 0.1) + (8 × 0.01) or as (9 × 100,000,000) + (5 × 1,000,000) + (1 × 100,000) + (5 × 10,000) + (8 × 10) + (9 × 1) + (8 × 0.01) or e.g., 905,150,089.08 as (9 × 100,000,000) + (0 × 10,000,000) + (5 × 1,000,000) + (1 × 100,000) + (5 × 10,000) + (0 × 1,000) + (0 × 100) + (8 × 10) + (9 × 1) + (0 × ) + (8 × ) or as (9 × 100,000,000) + (5 × 1,000,000) + (1 × 100,000) + (5 × 10,000) + (8 × 10) + (9 × 1) + (8 × )).
 Expanded notation is written following the order of place value.
 Expanded notation to represent the value of a digit(s) within a number
 Multiple representations of various forms of a number
 Equivalent relationships between place value of decimals through the hundredths (e.g., 0.2 is equivalent to 20 hundredths).
Note(s):
 Grade Level(s):
 Grade 3 composed and decomposed numbers up to 100,000 as a sum of so many ten thousands, so many thousands, so many hundreds, so many tens, and so many ones using objects, pictorial models, and numbers, including expanded notation as appropriate.
 Grade 4 introduces the millions and billions period.
 Grade 4 introduces representing the value of a decimal to the hundredths using expanded notation and numerals.
 Grade 5 will represent the value of the digit in decimals through the thousandths using expanded notation and numerals.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.2. Interpret the relationships between the different representations of numbers.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.

4.2C 
Compare and order whole numbers to 1,000,000,000 and represent comparisons using the symbols >, <, or =.
Supporting Standard

Compare, Order
WHOLE NUMBERS TO 1,000,000,000
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
 Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
 Place value – the value of a digit as determined by its location in a number, such as ones, tens, hundreds, one thousands, ten thousands, etc.
 Compare numbers – to consider the value of two numbers to determine which number is greater or less or if the numbers are equal in value
 Relative magnitude of a number describes the size of a number and its relationship to another number.
 Compare two numbers using place value charts.
 Compare digits in the same place value positions beginning with the greatest value.
 If these digits are the same, continue to the next smallest place until the digits are different.
 Numbers that have common digits but are not equal in value (different place values)
 Numbers that have a different number of digits
 Compare two numbers using a number line.
 Number lines (horizontal/vertical)
 Proportionally scaled number lines (predetermined intervals with at least two labeled numbers)
 Open number lines (no marked intervals)
 Order numbers – to arrange a set of numbers based on their numerical value
 A set of numbers can be compared in pairs in the process of ordering numbers.
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
 Order a set of numbers on a number line.
 Order a set of numbers on an open number line.
 Quantifying descriptors (e.g., between two given numbers, greatest/least, ascending/descending, tallest/shortest, warmest/coldest, fastest/slowest, longest/shortest, heaviest/lightest, closest/farthest, oldest/youngest, etc.)
Represent
COMPARISONS OF WHOLE NUMBERS TO 1,000,000,000 USING THE SYMBOLS >, <, OR =
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
 Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
 Comparative language and comparison symbols
 Inequality words and symbols
 Greater than (>)
 Less than (<)
 Equality words and symbol
Note(s):
 Grade Level(s):
 Grade 3 compared and ordered whole numbers up to 100,000 and represented comparisons using the symbols >, <, or =.
 Grade 5 will compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.

4.2E 
Represent decimals, including tenths and hundredths, using concrete and visual models and money.
Supporting Standard

Represent
DECIMALS, INCLUDING TENTHS AND HUNDREDTHS, USING CONCRETE AND VISUAL MODELS AND MONEY
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
 Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
 Decimals (less than or greater than one to the tenths and hundredths)
 Decimal number – a number in the base10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
 Various concrete and visual models
 Number lines, decimal disks, decimal grids, base10 blocks, money, etc.
Note(s):
 Grade Level(s):
 Previous grade levels used the decimal point in money only.
 Grade 4 introduces representing decimals, including tenths and hundredths, using concrete and visual models and money.
 Grade 5 will represent the value of the digit in decimals through the thousandths using expanded notation and numerals.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.2. Interpret the relationships between the different representations of numbers.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.

4.2F 
Compare and order decimals using concrete and visual models to the hundredths.
Supporting Standard

Compare, Order
DECIMALS USING CONCRETE AND VISUAL MODELS TO THE HUNDREDTHS
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
 Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
 Decimals (less than or greater than one to the tenths and hundredths)
 Decimal number – a number in the base10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
 Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, ten thousands, etc.
 Comparative language and comparison symbols
 Inequality words and symbols
 Greater than (>)
 Less than (<)
 Equality words and symbol
 Compare numbers – to consider the value of two numbers to determine which number is greater or less or if the numbers are equal in value
 Relative magnitude of a number describes the size of a number and its relationship to another number.
 Compare two decimals using place value charts.
 Compare digits in the same place value position beginning with the greatest place value.
 If these digits are the same, continue to the next smallest place until the digits are different.
 Numbers that have common digits but are not equal in value (different place values)
 Numbers that have a different number of digits
 Compare two decimals with various concrete and visual models.
 Number lines, decimal disks, decimal grids, base10 blocks, money, etc.
 Order numbers – to arrange a set of numbers based on their numerical value
 Order three or more decimals with various concrete and visual models.
 Quantifying descriptors (e.g., between two given numbers, greatest/least, ascending/descending, tallest/shortest, warmest/coldest, fastest/slowest, longest/shortest, heaviest/lightest, closest/farthest, oldest/youngest, etc.)
 Number lines, decimal disks, decimal grids, base10 blocks, money, etc.
Note(s):
 Grade Level(s):
 Grade 4 introduces comparing and ordering decimals using concrete and visual models to the hundredths.
 Grade 5 will compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.

4.2H 
Determine the corresponding decimal to the tenths or hundredths place of a specified point on a number line.
Supporting Standard

Determine
THE CORRESPONDING DECIMAL TO THE TENTHS OR HUNDREDTHS PLACE OF A SPECIFIED POINT ON A NUMBER LINE
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
 Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
 Decimals (less than or greater than one to the tenths and hundredths)
 Decimal number – a number in the base10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
 All decimals, to the tenths or hundredths place, can be located as a specified point on a number line.
 Characteristics of a number line
 A number line begins as a line with predetermined intervals (or tick marks) with positions/numbers labeled.
 A minimum of two positions/numbers should be labeled.
 Numbers on a number line represent the distance from zero.
 The distance between the tick marks is counted rather than the tick marks themselves.
 The placement of the labeled positions/numbers on a number line determines the scale of the number line.
 Intervals between position/numbers are proportional.
 When reasoning on a number line, the position of zero may or may not be placed.
 When working with larger numbers, a number line without the constraint of distance from zero allows the ability to “zoomin” on the relevant section of the number line.
 Number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
 Characteristics of an open number line
 An open number line begins as a line with no intervals (or tick marks) and no positions/numbers labeled.
 Numbers/positions are placed on the empty number line only as they are needed.
 When reasoning on an open number line, the position of zero is often not placed.
 When working with larger numbers, an open number line without the constraint of distance from zero allows the ability to “zoomin” on the relevant section of the number line.
 The placement of the first two numbers on an open number line determines the scale of the number line.
 Once the scale of the number line has been established by the placement of the first two numbers, intervals between additional numbers placed are approximately proportional.
 The differences between numbers are approximated by the distance between the positions on the number line.
 Open number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
 Landmark (or anchor) numbers may be placed on the open number line to help locate other numbers.
 Purpose of open number line
 Open number lines allow for the consideration of the magnitude of numbers and the placevalue relationships among numbers when locating a given whole number
 Number lines representing values less than one to the tenths place
 Number lines representing values greater than one to the tenths place
 Number lines representing values less than one to the hundredths place
 Number lines representing values greater than one to the hundredths place
 Number lines representing values between tick marks
 Relationship between tenths and hundredths
Note(s):
 Grade Level(s):
 Grade 3 represented a number on a number line as being between two consecutive multiples of 10; 100; 1,000; or 10,000 and use words to describe relative size of numbers in order to round whole numbers.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.2. Interpret the relationships between the different representations of numbers.

4.3 
Number and operations. The student applies mathematical process standards to represent and generate fractions to solve problems. The student is expected to:


4.3G 
Represent fractions and decimals to the tenths or hundredths as distances from zero on a number line.
Supporting Standard

Represent
DECIMALS TO THE TENTHS OR HUNDREDTHS AS DISTANCES FROM ZERO ON A NUMBER LINE
Including, but not limited to:
 Decimals (less than or greater than one to the tenths and hundredths)
 Decimal number – a number in the base10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
 Characteristics of a number line
 A number line begins as a line with predetermined intervals (or tick marks) with positions/numbers labeled.
 A minimum of two positions/numbers should be labeled.
 Numbers on a number line represent the distance from zero.
 The distance between the tick marks is counted rather than the tick marks themselves.
 The placement of the labeled positions/numbers on a number line determines the scale of the number line.
 Intervals between position/numbers are proportional.
 When reasoning on a number line, the position of zero may or may not be placed.
 When working with larger numbers, a number line without the constraint of distance from zero allows the ability to “zoomin” on the relevant section of the number line.
 Number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
 Decimals to the tenths or hundredths as distances from zero on a number line
 Relationship between a decimal represented using a strip diagram to a decimal represented on a number line
 Strip diagram – a linear model used to illustrate number relationships
 Decimals as distances from zero on a number line greater than 1
 Point on a number line read as the number of whole units from zero and the decimal amount of the next whole unit
 Number line beginning with a number other than zero
 Distance from zero to first marked increment is assumed even when not visible on the number line.
 Relationship between decimals as distances from zero on a number line to decimal measurements as distances from zero on a metric ruler, meter stick, or measuring tape
 Measuring a specific length using a starting point other than zero on a metric ruler, meter stick, or measuring tape
 Distance from zero to first marked increment not counted
 Length determined by number of whole units and the fractional amount of the next whole unit
Note(s):
 Grade Level(s):
 Grade 3 represented fractions greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 using concrete objects and pictorial models, including strip diagrams and number lines.
 Grade 3 determined the corresponding fraction greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 given a specified point on a number line.
 Grade 5 will represent the value of the digit in decimals through the thousandths using expanded notation and numerals.
 Grade 6 will identify a number, its opposite, and its absolute value.
 Grade 6 will locate, compare, and order integers and rational numbers using a number line.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.2. Interpret the relationships between the different representations of numbers.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
